After fitting an LCA model to one sample, is it possible to get Mplus to compute posterior probabilities on a new sample using the parameter estimates from the original sample and the raw data from the new sample? What would be the best way of doing this and getting accurate results?

The way to do this is to fix all parameters to the model estimated values from the first sample. When you run the analysis with the second data set, be sure the model has no free parameters. Ask for CPROBABILITIES in the SAVEDATA command. You are making the assumption that the second sample comes from the same population as the first sample.